r.resamp.bspline performs a bilinear/bicubic spline interpolation with Tykhonov regularization. The input is a raster surface map, e.g. elevation, temperature, precipitation etc. Output is a raster map. Optionally, only input NULL cells are interpolated, useful to fill NULL cells, an alternative to r.fillnulls. Using the -n flag to only interpolate NULL cells will considerably speed up the module.

The input raster map is read at its native resolution, the output raster map will be produced for the current computational region set with g.region. Any MASK will be respected, masked values will be treated as NULL cells in both the input and the output map.

Spline step values ew_step for the east-west direction and ns_step for the north-south direction should not be smaller than the east-west and north-south resolutions of the input map. For a raster map without NULL cells, 1 * resolution can be used, but check for undershoots and overshoots. For very large areas with missing values (NULL cells), larger spline step values may be required, but most of the time the defaults (1.5 x resolution) should be fine.

The Tykhonov regularization parameter (lambda) acts to smooth the interpolation. With a small lambda, the interpolated surface closely follows observation points; a larger value will produce a smoother interpolation. Reasonable values are 0.0001, 0.001, 0.005, 0.01, 0.02, 0.05, 0.1 (needs more testing). For seamless NULL cell interpolation, a small value is required and default is set to 0.005.

From a theoretical perspective, the interpolating procedure takes place in two parts: the first is an estimate of the linear coefficients of a spline function; these are derived from the observation points using a least squares regression; the second is the computation of the interpolated surface (or interpolated vector points). As used here, the splines are 2D piece-wise non-zero polynomial functions calculated within a limited 2D area. The length of each spline step is defined by ew_step for the east-west direction and ns_step for the north-south direction. For optimal performance, the spline step values should be no less than the east-west and north-south resolutions of the input map. Each non-NULL cell observation is modeled as a linear function of the non-zero splines in the area around the observation. The least squares regression predicts the the coefficients of these linear functions. Regularization avoids the need to have one one observation and one coefficient for each spline (in order to avoid instability).

A cross validation "leave-one-out" analysis is available to help to determine the optimal lambda value that produces an interpolation that best fits the original observation data. The more points used for cross-validation, the longer the time needed for computation. Empirical testing indicates a threshold of a maximum of 100 points is recommended. Note that cross validation can run very slowly if more than 100 observations are used. The cross-validation output reports mean and rms of the residuals from the true point value and the estimated from the interpolation for a fixed series of lambda values. No vector nor raster output will be created when cross-validation is selected.

In this example, the SRTM elevation map in the North Carolina sample dataset location is filtered for outlier elevation values; missing pixels are then re-interpolated to obtain a complete elevation map: